The weak field approximation for general relativity with and without

torsion and its applications

Miriam Loois

Supervisor: Frank Witte

July 13, 2006

Bachelor Thesis Faculty of Science/Department of Physics and Astronomy

The weak field approximation for

general relativity with and without

torsion and its applications

Miriam Loois

Bachelor Thesis

Supervisor: Frank Witte

Faculty of Science/Department of Physics and Astronomy

July 13, 2006

Abstract

First we give an overview of general relativity, the vielbein method, geomet-ric algebra, the weak field approximation and gravitational waves. We set upa weak field approximation for the vielbein method and use this to describegravitational waves in geometric algebra. Further we look at torsion and it’sinfluence on the geodesic (deviation) equation. In the particular case wheretorsion is related to the spin density, we see that geodesics remain the same,but the separation vector changes due to torsion. We set up a weak field ap-proximation for general relativity with torsion. Finally we compute the spindensities for a particle in a superposition of spin up and spin down, and solvethe differential equations for the separation vector and we explain that torsioncan be the reason that we never see superpositions in the classical limit.

Contents

1 Introduction 2

2 General Relativity 32.1 Theory of General Relativity . . . . . . . . . . . . . . . . . . . . 32.2 General Relativity with Torsion . . . . . . . . . . . . . . . . . . . 5

2.2.1 Influence of torsion on geodesic (deviation) equation . . . 52.2.2 Torsion and spin density . . . . . . . . . . . . . . . . . . . 6

3 Alternative Approaches 83.1 Coordinate versus Coordinate Free Bases . . . . . . . . . . . . . 83.2 Geometric algebra approach . . . . . . . . . . . . . . . . . . . . . 9

4 Weak Field Approximation 114.1 Weak Field approximation using a coordinate basis . . . . . . . . 114.2 Weak Field approximation using a coordinate free basis . . . . . 114.3 Weak field approximation for general relativity with torsion . . . 12

5 Gravitational Radiation versus Torsion 145.1 Particles in a gravitational wave without torsion . . . . . . . . . 145.2 Source of Gravitational radiation . . . . . . . . . . . . . . . . . . 185.3 Torsional radiation . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6 Discussion 20

7 Acknowledgements 20

A Maple worksheet for computing spin densities 22

1

1 Introduction

In general relativity, most of the time one assumes that the connection is sym-metric, and torsion, the antisymmetric part, is zero. One can ask if this isfundamentally right, if it is a good approximation, or that it is a gauge. In thelast case, one could make another gauge with the same physical results. Theaim of this paper is to look at general relativity with and without torsion atdifferent ways. Especially the weak field limit is a good way to do this. Becauseof the simplifications of the weak field limit, it is possible to compute geodesicsand geodesic deviations for simplistic physical systems (which is very difficultwithout this approximation). Having done this, the results can be compared,and we can form ideas about what torsion actually is.

The first chapter is divided in two sections. In the first subsection we givean overview of the most important results of general relativity, and the thingsthat are important for further calculations. This can be skipped by peoplethat are already familiar with general relativity. In the second one, we give anoverview of the effect of torsion on general relativity. In the second chapter weintroduce two different approaches to general relativity, namely a coordinatefree basis, and geometric algebra. In the third chapter we look at the weak fieldlimit, and how the important quantities can be simplified in this limit. We alsoset up a weak field limit for general relativity with torsion. This weak field limitwill be used in the last chapter to compute the influence of the spin density onthe geodesic deviation equation. We also give an other application of the weakfield limit, namely gravitational radiation and we compare the two with eachother. Finally, in the discussion we look back at the found results and we givea suggestion for further research.

2

2 General Relativity

The first part of this chapter is based on the lecture notes of Sean Carrol [1],and is about general relativity without torsion. The second part is based onan article of Watanabe [9], and describes the influence of torsion and spin ongeneral relativity.

2.1 Theory of General Relativity

In special relativity it is assumed that we live in a Minkowski space. In sucha space, spacetime is flat, and every point in space is represented by one timeand three space coordinates, xµ, where µ = 0 represents the time coordinateand µ = 1, 2, 3 the space coordinates. If we multiply these kind of vectors (withthemselves or with others), we must make use of a metric η , which can berepresented by a matrix.

ηµν =

−1 0 0 00 1 0 00 0 1 00 0 0 1

The dot product between two vectors, making use of the Einstein summationconvention, is then defined as

V ·W = ηµνVµW ν (1)

In general relativity space is no longer flat, but space can be curved by en-ergy and momentum. Curved space is represented by a manifold. For everypoint on the manifold there is a tangent space. The coordinate system of anobserver at a certain point P lies within the tangent space of P. There are fourbasis vectors, e(µ) = ∂µ. In general relativity, vectors can only be compared ifthey are in the same tangent space. The metric is no longer the same as in aMinkowski space, but is now denoted by gµν , and can vary through space.

One of the ideas of general relativity is that the laws of nature should beindependent of the coordinate system you choose. Therefore, we want to definea coordinate invariant derivative, the covariant derivative, that satisfies the fol-lowing conditions:

linearity: ∇(T + S) = ∇T +∇SLeinbiz product rule: ∇(T

⊗S) = (∇T )

⊗S + T

⊗(∇S)

Therefore we have to write it as a partial derivative plus a correction, knownas the connection coefficients.

∇µV ν = ∂µVν + ΓνµλV

λ (2)

To get a unique connection coefficient, we introduce two conditions:

3

Torsion free: Tλµν = 12 (Γλµν − Γλνµ) = 0

metric compatibility: ∇ρgµν = 0

Now the connection coefficients are related to the metric via:

Γσµν =12gσρ(∂µgνρ + ∂νgρµ − ∂ρgµν) (3)

The connection coefficients are zero in flat space, because the Minkowski metricis constant. We will call this specific expression for the connection the Christof-fel connection.

In flat space we can move a vector while keeping its direction and lengthconstant. The equivalent in curved space is parallel transport. But if youparallel transport a tensor along curved space, it is path-dependent, while inflat space it’s not. In flat space the condition for keeping a tensor constant alonga curve xµ(λ) is dT

dλ = 0 In curved space this can be generalized to:

D

dλT ≡ dxµ

dλ∇µT = 0 (4)

This leads also to the equation for geodesics, which are roughly the orbits thatfree particles describe on a curved space:

d2xµ

dλ2+ Γµρσ

dxρ

dλ

dxσ

dλ= 0 (5)

The connection coefficients can be used to describe the curvature of space.First we have to define the Riemann Curvature tensor, the Ricci tensor and theRicci scalar:

Riemann curvature tensor : Rρσµν = ∂µΓρνσ − ∂νΓρµσ+ΓρµλΓλνσ − ΓρνλΓλµσ (6)

Ricci tensor : Rµν = Rλµλν (7)Ricci scalar : R = Rµµ = gµνRµν (8)

A positive Ricci scalar means space curves away in the same way in everydirection (for example a sphere), while a negative Ricci scalar means the cur-vature in one direction is opposite to that of an other direction (a so calledsaddle point). One can always choose a coordinate system where the Christoffelconnection is zero at a certain point, but one can not do such a thing for theRiemann curvature tensor.

4

The Einstein tensor is defined as:

Gµν = Rµν − 12Rgµν (9)

Einstein found a relation between the curvature of space and the distributionof energy and momentum in the universe, also known as the Einstein equation:

Gµν = 8πGTµν (10)

Where Tµν is the flux of the µth component of momentum in the νth directionand G is the gravitational constant.

Imagine we have a collection of geodesics γs(t) and points xµ(s, t) on themanifold. We can define a separation vector S, that points from geodesic togeodesic and a tangent vector T , tangent to a geodesic:

Sµ =∂xµ

∂s(11)

Tµ =∂xµ

∂t(12)

In flat space, the separation vector is constant, because the distant between twogeodesics remains constant. In curved space the distant between two geodesicscan vary, and therefor the separation vector is not constant.There is also an equation for the separation vector, known as the geodesicdeviation equation:

D2

dλ2Sµ = RµνρσT

νT ρSσ (13)

Two observers can only measure their relative position, not the geodesicsthey follow. When two observers experience a force, for example a gravitationalforce, their relative position will change. Therefore the Riemann curvaturetensor can be interpreted as a force.

2.2 General Relativity with Torsion

2.2.1 Influence of torsion on geodesic (deviation) equation

If we allow torsion the connection changes to:

Γλµν = Γλµν + Tλµν − T λµν − T λ

νµ (14)

Here Γλµν is the Christoffel connection (3). We see that the last two termsalso contribute to the symmetric part of the connection. Only if Tλµν is fully an-tisymmetric, the symmetric part vanishes. Now let’s take a look at the geodesicand geodesic deviation equation.

5

d2xµ

dλ2+ Γµρσ

dxρ

dλ

dxσ

dλ= 0 (15)

D2

dt2Sµ = RµνρσT

νT ρSσ (16)

We see that in both equations the connection and the Riemann tensor occurin pairs. Of the connection pair Γµρσ + Γµσρ only the symmetric part survives,so the last two torsion terms do effect the geodesic equation, but the first onedoes not. For the Riemann pair we get:

Rρσµν +Rρµσν = Rρσµν + Rρµσν

+∂µT ρνσ + ∂σTρνµ − ∂µ(T ρ

νσ + T ρσν )

−∂σ(T ρνµ + T ρ

µν ) + 2∂ν(T ρνσ + T ρ

σν ) (17)

We see that in this case all three terms affect the geodesic deviation equa-tion. So if torsion is totally antisymmetric, the geodesics don’t change, but theseparation vectors do.

2.2.2 Torsion and spin density

According to an article of T. Watanabe [9] torsion is related to the spin density:

τλµν = Tλµν + 2eλ[µTλνλ] (18)

Here eλµ is a 4 × 4 matrix that is not important now, but will be discussed inchapter 3. Watanabe finds an equation for the spin density:

τλµν =−πGi

2ψ{γλ, [γµ, γν ]}ψ (19)

The curly brackets denote the anti-commutator and the square brackets thenormal commutator. The gamma matrices are given by:

γ0 =

−1 0 0 00 1 0 00 0 1 00 0 0 1

, γ1 =

0 0 0 10 0 1 00 −1 0 0−1 0 0 0

(20)

γ2 =

0 0 0 −i0 0 i 00 i 0 0−i 0 0 0

, γ3 =

0 0 1 00 0 0 −1−1 0 0 00 1 0 0

(21)

Further, ψ = ψ†γ0 and γµ = ηµνγν .

6

The first thing we see is that the spin density is totally antisymmetric. There-fore torsion is also totally antisymmetric and equal to the spin density, does notaffect the geodesic equation, but does affect the geodesic deviation equation. Sowe can write:

Γλµν = Γλµν + τλµν (22)

An older paper of F.W. Hehl [7] also gives an expression for the spin den-sity, but when we compared the results to the ones obtained in the paper ofWatanabe, there seemed to be a difference.

7

3 Alternative Approaches

In this section, we give an overview of two different approaches to general rela-tivity. They are both not based on different physics, but try to get new resultsby writing things in a different formalism. The first part is based on the lecturenotes of Sean Carrol, the second part on the lecture notes about geometric andspacetime algebra of Frank Witte [3].

3.1 Coordinate versus Coordinate Free Bases

The most obvious way to choose a basis is a coordinate basis. Every basis vectorat a point p is then a partial derivative with respect to the coordinates of p,e(µ) = ∂µ. In this case, a vector V can be written as V = V µe(µ). Differentbases and vector components are related to each other by a general coordinatetransformation:

∂µ′ =∂xµ

∂xµ′∂µ (23)

V µ′

=∂xµ

′

∂xµV µ (24)

But we can also choose an arbitrary basis, e(a), that is not based on acoordinate system, as long as the basis vectors are orthonormal. These basesare related to each other by:

e(µ) = eaµe(a) (25)

The 4 × 4 matrix eaµ is often called a vierbein and can also be used to switchfrom coordinate based vector components to coordinate free components andvice versa via:

V a = eaµVµ (26)

By a Local Lorentz Transformation one coordinate free basis can be ex-pressed into another:

e(a′) = Λ aa′ e(a) (27)

A tensor with Latin en Greek indices (where the Greek indices are related to acoordinate basis and the Latin ones are not) now transforms as:

T a′µ′

b′ν′ = Λa′a

∂xµ′

∂xµΛ bb′∂xν

∂xν′T aµbν (28)

The covariant derivative also acts different on a tensor with Latin indices:

∇µXab = ∂µX

ab + ω a

µ cXcb − ω c

µ bXac (29)

The spin connection ω aµ b can be expressed in terms of the (normal) connection

and the vierbeins by:ω aµ b = eaνe

λbΓνµλ − eλb ∂µeaλ (30)

8

3.2 Geometric algebra approach

In geometric, or Clifford algebra we use the Clifford product that satisfied thefollowing rules for vectors a and constants k:

a(bc) = (ab)c (31)a(b + c) = ab + bc (32)

ka = ak (33)

We assume that a2 is a number and it represents the length of a vector a.Further, we split the product in a symmetric and an antisymmetric part, whichwe will call the dot product and the wedge product:

a · b =12

(ab− ba) (34)

a ∧ b =12

(ab + ba) (35)

The wedge product is the four dimensional variant of the outer product. Nowthe geometric product between to vectors can be written as:

ab = a · b + a ∧ b (36)

Just as in special relativity we use four basis vectors that satisfy the condi-tions:

eµ · eν = 0, µ 6= ν (37)e2

0 = −1, e21 = e2

2 = e23 = 1 (38)

Further we have six bivectors:

J1 = e2e3, J2 = e3e1, J3 = e1e2 (39)K1 = e0e1, K2 = e0e2, K3 = e0e3 (40)

Four trivectors and a pseudoscalar i that has the nice property that i2 = −1:

T0 = e1e2e3, Ti = e0Ji (41)i = e0e1e2e3 (42)

Every multivector M is the sum of coefficients times a vector, a bi-, or trivec-tor or the pseudoscalar. In general, the dot product between two multivectorsM and N is no longer just MN + NM as it is for a vector. For example, fortwo bivector r and s the dot product is rs− sr.

The geodesic equation in geometric algebra becomes:

v = −ω(v) · v (43)

9

This looks very simple, but the problem is to find ω. First of all, ω(v) is acontraction of ωµvµ. The components can be computed by:

ωµ = ωα βµ (eα ∧ eβ) (44)

The components ωα βµ are the same as the components of the spin connection

computed in the previous subsection.

10

4 Weak Field Approximation

In the weak field limit we look at a flat space with a small perturbation. In thefirst part of this section we summarize the results for a coordinate basis. Thisis based on the lecture notes of Sean Carroll. In the second part, we set up aweak field approximation for a coordinate free basis.

4.1 Weak Field approximation using a coordinate basis

In the weak field approximation we look at the Minkowski metric with a smallperturbation.

gµν = ηµν + hµν (45)

With it’s inverse gµν = ηµν − hµν . We assume |hµν | and |∂ρhµν | much smallerthan unity. In this approximation we neglect all terms of order h2 and higher.Because the solutions of the Einstein equation are not unique, we choose towork in a harmonic coordinate system. In this system the gauge condition isgµνΓλµν = 0. In the weak field approximation this gauge reduces to ∂λhλµ =12∂µh.

In the following table the general quantaties are compared with the quan-taties in the weak field limit.

General Weak field limitΓσµν = 1

2gσρ(∂µgνρ + ∂νgρµ − ∂ρgµν) Γσµν = 1

2ησρ(∂µhνρ + ∂νhρµ − ∂ρhµν)

Rρσµν = ∂µΓρνσ − ∂νΓρµσ + ΓρµλΓλνσ − ΓρνλΓλµσ Rρσµν = 12 (∂µ∂σhρν + ∂ν∂ρhσµ − ∂ν∂σhρµ − ∂µ∂ρhσν)

Rµν = Rλµλν Rµν = 12 (∂σ∂νhσµ + ∂σ∂µh

σν − ∂µ∂νh−2hµν)

R = gµνRµν R = ∂λ∂µhλµ −2h

Gµν = Rµν − 12Rgµν Gµν = Rµν − 1

2ηµνRGµν = 8πGTµν 2hµν = −16πGTµν

Where h = ηµνhµν , 2 = ηµν∂µ∂ν and hµν = hµν − 12ηµνh

Now we can also simplify the geodesic deviation equation (13) in the limit ofslow moving particles. Because the time dilatation for slow moving particles isnegligible, we set λ = t and we choose T ν = (1, 0, 0, 0), because the correctionson T are of order h and the Riemann tensor is already of order h. Now thegeodesic deviation equation becomes:

d2Sµ

dt2= Rµ00σS

σ (46)

4.2 Weak Field approximation using a coordinate free ba-sis

The metric can be expressed in terms of the vierbeins [1]:

gµν = eaµebνηab (47)

11

We want to make an approximation for the vierbeins, such that the metricis again the Minkowski metric plus a small perturbation. We assume metriccompatibility, but we don’t assume torsion to be zero. We write the vierbeinsas the identity matrix plus a small perturbation:

eaµ = δaµ +12haµ (48)

Again |hµν | and |∂ρhµν | are much smaller than one and the inverse of eaµ iseµa = δµa − 1

2hµa. Now we can use this approximation to compute the metric,

the connection and the spin connection up to the first order in h:

gµν = eaµebνηab = (δaµ +

12haµ)(δbν +

12hbν)ηab = ηµν +

12

(hµν + hνµ) (49)

We see that gµν automatically becomes symmetric, so nothing stops us fromchoosing a nonsymmetric perturbation on eµa . Making use of

∂µgνρ = ηab(eaν∂µebρ + ebρ∂µe

aν) =

12∂µ(hνρ + hρν) (50)

we find for the Christoffel symbol:

Γσµν =14ησρ[∂µ(hνρ + hρν) + ∂ν(hρµ + hµρ)− ∂ρ(hµν + hνµ)] (51)

This is exactly what we would expect. Further,

ω aµ b = eaνe

λbΓνµλ − eλb ∂µeaλ

= Γaµb −12∂µh

ab

=14ηaρ(∂µ(hbρ + hρb) + ∂b(hµρ + hρµ)− ∂ρ(hµb + hbµ)

−12ηaρ∂µhρb (52)

When we assume hµν to be symmetric, the spin connection reduces to:

ω aµ b =

12ηaρ(∂bhµρ − ∂ρhµb) (53)

4.3 Weak field approximation for general relativity withtorsion

Because general relativity without torsion seems to work pretty good, we assumethe spin density to be very small and we set up a weak field approximation, wherethe Christoffel connection and torsion are both of order h, and we neglect orderh2. Now we compute the Riemann tensor, because we need it for the geodesicdeviation equation:

12

Rρσµν = ∂µΓρνσ − ∂νΓρµσ = Rρσµν + ∂µτρνσ − ∂ντρµσ (54)

Rρσµν is the Riemann curvature tensor without torsion, computed in section4.1. In particular, we need the Riemann tensor with indices σ and µ equal tozero. Because the spin density is totally antisymmetric, τρ00 necessarily has tobe zero, and we get:

Rρ00ν = Rρ00ν + ∂0τρν0 (55)

Now lets take a look at the situation where the metric is totally flat, sothe only perturbation comes from the spin density. This leads to the followingequation for the separation vector:

d2Sµ

dt2= ∂0τ

µσ0S

σ (56)

13

5 Gravitational Radiation versus Torsion

In this section we look at two applications of the weak field limit, namely gravi-tational radiation and torsion. The first two sections, expect for the part aboutgeodesics, are based on the lecture notes of Sean Carroll.

5.1 Particles in a gravitational wave without torsion

The geodesic deviation equation

We are back at the case where there is no torsion and the metric is symmetric.We are going to look at a wave-perturbation of space in vacuum. We saysomething produces radiation if it produces an perturbation of the metric ofthe following form:

hµν = Cµνeikσx

σ

(57)

Because we are in vacuum 2hµν = 0, and therefore we must have kσkσ = 0.The harmonic gauge leads to kµCµν = 0, and if we choose our wave to travelin the z-direction (k = (ω, 0, 0, ω)) it has been shown that al solutions can bewritten in the following form:

Cµν =

0 0 0 00 C+ C× 00 C× −C+ 00 0 0 0

A transverse traceless gauge is used and in this gauge we can replace all hµνby hµν . For slow moving particles we have:

d2Sµ

dt2= Rµ00σS

σ (58)

Where Rµ00σ = 12∂t∂th

µσ. Now we are going to look at the cases were C×

equals zero and C+ is not and vice versa, which we will call the plus resp. crosspolarization. This leads to the following set of differential equations and solu-tions op to the first order in h:

Plus polarization Cross polarization∂2Sx

∂t2 = − 12ω

2C+eiω(z−t)Sx ∂2Sx

∂t2 = − 12ω

2C×eiω(z−t)Sy∂2Sy

∂t2 = 12ω

2C+eiω(z−t)Sy ∂2Sy

∂t2 = − 12ω

2C×eiω(z−t)Sx

Sx = (1 + 12C+(eiω(z−t) − eiω(z))Sx(0) Sx = Sx(0) + 1

2C×(eiω(z−t) − eiω(z))Sy(0)Sy = (1− 1

2C+(eiω(z−t) − eiω(z)))Sy(0) Sy = Sy(0) + 12C×(eiω(z−t) − eiω(z))Sx(0)

If you start with a ring of particles in the x-y plane and the gravitationalwave is plus polarized, they will start to move in the shape of a plus (figure 1),when the wave is cross polarized, in the form of a cross (figure 2).

14

Figure 1: Plus polarization

Figure 2: Cross polarization

The geodesic equation

Now we want to calculate the geodesics that particles will follow, not onlythe difference between the particles. Therefore we look at the geodesic equation(5). For the plus polarization, the only nonzero terms of the connection are:

Γxxz = Γxzx =12∂zh

xx, Γyyz = Γyzy =

12∂zh

yy

Γxxt = Γxtx =12∂th

xx, Γyyt = Γyty =

12∂th

yy (59)

Γt xx =12∂thxx, Γt yy =

12∂thyy

Γzxx = −12∂zhxx, Γzyy = −1

2∂zhyy

15

Now we get the following set of differential equations:

d2x

dλ2= iωC+e

iω(z−t)(dx

dλ

dz

dλ− dx

dλ

dt

dλ) (60)

d2y

dλ2= −iωC+e

iω(z−t)(dy

dλ

dz

dλ− dy

dλ

dt

dλ) (61)

d2z

dλ2= iωC+e

iω(z−t)((dy

dλ)2 − (

dx

dλ)2) (62)

d2t

dλ2= iωC+e

iω(z−t)((dy

dλ)2 − (

dx

dλ)2) (63)

The first thing we see is that the second derivatives of z and t are equal, so

z − t = z0 − t0 + (dz

dλ(0)− dt

dλ(0))λ (64)

When we go back to slow moving particles, we write:

dxν

dλ= (1, 0, 0, 0) + ~p(λ) (65)

and we neglect p2. Then the equations for z and t vanish and the other tworeduce to:

d2x

dt2= −iωC+e

iω(z−t) dxdt

(66)

d2y

dt2= iωC+e

iω(z−t) dydt

(67)

These two equations can easily be solved, and the solutions are:

x(t) = x0 + (1− C+eiωz)v0xt+ v0x

C+

ωieiω(z−t) (68)

y(t) = y0 + (1 + C+eiωz)v0yt− v0y

C+

ωieiω(z−t) (69)

Following the same procedure, we find for the cross polarization that:

x(t) = x0 + (v0x − C×eiωzv0y)t+ v0yC+

ωieiω(z−t) (70)

y(t) = y0 + (v0y − C×eiωzv0x)t+ v0xC+

ωieiω(z−t) (71)

Where v0x = dxdt (0) and v0y = dy

dt (0). We see that the particles move ina straight line with a small sine or cosine perturbation perpendicular to thedirection of propagation of the wave, as we would aspect from the solution forthe separation vector. We also see that their initial velocity changes a little bitdue to the gravitational wave. In figure 3 some geodesics with the same initialspeed, but different initial velocity are plot.

16

Figure 3: Geodesics of particles with the same initial velocity but different initialposition due to gravitational radiation

Because of (64) you can also solve the equations of motion exactly for x andy by using:

dx

dλ

dz

dλ− dx

dλ

dt

dλ=dx

dλ(dz

dλ− dt

dλ) =

dx

dλ(vz0 − vt0) (72)

When you have found the solutions for x and y, you can use them to find theequations for z and t. This is not difficult, but gets very messy. Important isthat the gravitational wave also affects the z and t direction, although not much.

Now we can also try to compute geodesics with geometric algebra. In theweak field approximation the spin connection reduces to:

ω aµ b =

12ηaρ(∂bhµρ − ∂ρhµb) (73)

For the cross polarization of a gravitational wave, hxy = hyx = C×eiω(z−t), weget the following components:

ω yx t =

12∂thxy = ω x

y t = ω tx y = ω t

y x (74)

ω yx z =

12∂zhxy = ω x

y z = −ω zx y = −ω z

y x (75)

And by raising and lowering with ηµν we get:

ωx ty = −1

2∂thxy = ωy t

x = ωx yt = ωy xt (76)

ωx zy =

12∂zhxy = ωy z

x = −ωx yz = −ωy x

z (77)

When we fill in these components in (43), we find the right differential equa-tions.

17

5.2 Source of Gravitational radiation

Now that we have computed the equations of motion for a particle in a grav-itational wave, we also want to know how it is generated and where it occursin nature. It can be shown that if we want to satisfy the condition for vacuum,2hµν = 0, and we are at a distant R from a source, the perturbation has to beof the following form:

hµν(t, ~x) =2G3R

d2qµνdt2

(tr) (78)

Where tr is the retarded time t−R and qµν is the quadrupole moment tensordefined as:

qµν(t) = 3∫xµxνT 00(t, ~x)d3x (79)

If you apply this for example to a binary star, you can compute that it emitsgravitational radiation.

5.3 Torsional radiation

A free particle with spin 12 can be described by the following wave equation:

ψ~p = u~pei(~p·~x−Ept) (80)

u~p,up =

10pz

Ep+mpx+ipyEp+m

, u~p,down =

01

px−ipyEp+m−pzEp+m

(81)

When we compute the spin densities for one particle with spin up or spindown, the exponentials cancel each other out, so there is no time-dependenceand it has no influence on the separation vector. Therefore we are going to lookat a particle in a superpositions of spin up and spin down with different energy.

First, let us look at the simplest case. A particle in a superposition of spinup and spin down with equal possibility, Ψ = 1√

2(ψup + ψdown),with an energy

difference E = Eup−Edown and both states with zero momentum. Now we getthe following spin densities:

τ txy = 0, τ txz = 2πG sin (Et), τ tyz = −2πG cos (Et), τxyz = 0 (82)

We have used the Maple worksheet that is included in Appendix A. Becausewe are in the weak field limit, the indices of the spin density can be raised and

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lowered by making use of ηµν and ηµν . Plugging everything in, this leads to thefollowing set of coupled differential equations:

d2Sx

dt2= (∂tτxzt)S

z = ∂t(−2πG sin (Et))Sz = −2πGE cos (Et)Sz (83)

d2Sy

dt2= (∂tτ

yzt)S

z = ∂t(2πG cos (Et))Sz = −2πGE sin (Et)Sz (84)

d2Sz

dt2= (∂tτzxt)S

x + (∂tτzyt)Sy (85)

= ∂t(2πG sin (Et))Sx + ∂t(−2πG cos (Et))Sy

= 2πGE(cos (Et)Sx + sin (Et)Sy)

The solutions, up to the first order of h are:

Sx(t) = Sx(0) +2πGE

(cos (Et)− 1)Sz(0) (86)

Sy(t) = Sy(0) +2πGE

sin (Et)Sz(0) (87)

Sz(t) = Sz(0)− 2πGE

((cos (Et)− 1)Sx(0) + sin (Et)Sy(0)) (88)

When the two particles have nonzero, but opposite momentum, pi = pup,i =−pdown,i, we obtain the same solutions for the separation vector, we only haveto replace cos (Et) by cos (Et− 2pixi) and sin (Et) by sin (Et− 2pixi). In thiscase the component τ txy of the spin density is no longer zero, but it is constant,so it does not contribute to the equation for the separation vector.If we set the x and y component of both particles equal to zero we also get thesame solution, but with the sine and cosine replaced by sin (Et+ (pz,down − pz,up)z)and cos (Et+ (pz,down − pz,up)z) respectively.

Figure 4 shows an example of the vector components Sx and Sy as a func-tion of x and y. The difference in orientation is exaggerated, because otherwiseit would be impossible to see.

If we compare the results of gravitational waves with those of particle in asuperposition of spin up and spin down, we see differences and similarities. Thesolutions of the separation vector are of the same type, a straight line witha small sine or cosine perturbation, and in both cases, the initial value of acomponent had influence on the other components. One of the differences isthat in the limit of slow moving particles, only the x- and y-component of theseparation vector are nonzero due tot gravitational waves, but spin also affectsthe z component. On the other hand, in the general case, gravitational wavescan also affect the z-component. Further, gravitational waves have influence ongeodesics, but spin density has not.

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Figure 4: Vector (Sx,Sy) as a function of x and y

6 Discussion

We have seen that gravitational and torsional radiation very much look alike.This could be an indication that the choice of setting torsion equal to zero isjust a gauge choice.We also have seen that particles that are in a superposition, experience a forcedue to torsion. This force becomes zero when the particle is in a single state.Therefore we can expect that this force tries to transform particles that are ina superposition into particles that are in a single state. This could be a reasonwhy in the classical limit we never notice the existence of superposition. On alarge timescale, an object will no longer be in a superposition, but has found astable, single state. A suggestion for further research could be to implement thefound torsional part of the connection in the Dirac equation. Maybe this couldgive us more information about the physics behind the collapse of wavefunction.

7 Acknowledgements

I would like to thank Frank Witte, who supervised this Bachelor Thesis, for allthe conversations we had and for answering all my questions. I also would liketo thank Erik Lascaris for giving linguistic advises and helpful Latex tips.

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References

[1] S. Carrol, Lecture Notes on General Relativity,1997,pancake.uchicago.edu/˜ carroll/notes

[2] I. Chakrabarty, Gravitational Waves: An Introduction, 1999

[3] F. Witte, Lecture Notes Geometric and Spacetime Algebra,www.phys.uu.nl/˜witte

[4] D. Hestenes, Gauge Theory Gravity with Geometric Calculus

[5] D. Hestenes, Spacetime Geometry with Geometric Calculus

[6] M. Francis et al., Geometric Algebra Techniques for General Relativity,Annals Phys. 311, 2004

[7] F.W. Hehl et al., General relativty with spin and torsion: Foundations andprospects, Rev. Mod. Phys., 48:393-416, 1976

[8] S. Jensen, General relativty with torsion: Extending Wald’s Chapter onCurvature, 2005

[9] T. Watanabe et al., General relativty with torsion, TOKAI-HEP/TH-04-09,2004

[10] J.T. Lunardi et al., Interacting spin 0 fields with torsion via Duffin-Kemmer-Petiau theory, Gen.Rel.Grav. 34, 2002

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A Maple worksheet for computing spin densi-ties

> restart:

> #Dirac matrices with upper indices#

> gu0:=matrix([[1,0,0,0],[0,1,0,0],[0,0,-1,0],[0,0,0,-1]]):

> gu1:=matrix([[0,0,0,1],[0,0,1,0],[0,-1,0,0],[-1,0,0,0]]):

> gu2:=matrix([[0,0,0,-I],[0,0,I,0],[0,I,0,0],[-I,0,0,0]]):

> gu3:=matrix([[0,0,1,0],[0,0,0,-1],[-1,0,0,0],[0,1,0,0]]):

> #Minkowski metric to raise and lower indices#

> eta:=matrix([[-1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]):

> #Dirac matrices with lower indices#

> gd0:=evalm(eta&*gu0):

> gd1:=evalm(eta&*gu1):

> gd2:=evalm(eta&*gu2):

> gd3:=evalm(eta&*gu3):

> #commutator of two dirac matrices multiplied with I/2#

> s12:=evalm(I/2*evalm(gd1&*gd2-gd2&*gd1)):

> s13:=evalm(I/2*evalm(gd1&*gd3-gd3&*gd1)):

> s23:=evalm(I/2*evalm(gd2&*gd3-gd3&*gd2)):

> s01:=evalm(I/2*evalm(gd1&*gd2-gd2&*gd1)):

> #anticommutator of a dirac matrix with> the commutator of two dirac> matrices#

> H123:=evalm(gu1&*s23+s23&*gu1):

> H013:=evalm(gu0&*s13+s13&*gu0):

> H023:=evalm(gu0&*s23+s23&*gu0):

> H012:=evalm(gu0&*s12+s12&*gu0):

> with(linalg):with(LinearAlgebra):

> #The spin up part of the wavefunction#> phi1:=matrix([[exp(I*(px1*x+py1*y+pz1*z-E1*t))],[0],[pz1/(E1+m1)*exp(> I*(px1*x+py1*y+pz1*z-E1*t))],[(px1+I*py1)/(E1+m1)*exp(I*(px1*x+py1*y+p> z1*z-E1*t))]]):

> #The spin down part#> phi2:=matrix([[0],[exp(I*(px2*x+py2*y+pz2*z-E2*t))],[(px2-I*py2)/(E2+> m2)*exp(I*(px2*x+py2*y+pz2*z-E2*t))],[-pz2/(E2+m2)*exp(I*(px2*x+py2*y+> pz2*z-E2*t))]]):

> #The superposition of spin up and down.> The factor one over the> square of two is added later#

> phi:=evalm(phi1+phi2):

> #phi dagger:#> phiherm:=htranspose(phi) assuming> E1::real, E2::real, px1::real, py1::real, pz1::real, px2::real,> py2::real, pz2::real, m1::real,> m2::real, t::real, x::real, y::real, z::real:

> #phi bar#

> phistreep:=evalm(phiherm&*gd0):

> #assumptions#

> px1:=-px2; py1:=-py2; pz1:=-pz2; E1:=E2+E;

px1 := −px2py1 := −py2pz1 := −pz2

E1 := E2 + E

> #We have to multiply with -pi*G and> divide by a factor two because> the superposition of the wavefunctions was not normalized. Nowwe compute the spin density components and their derivatives withrespect to t#

> t012:=simplify(evalm(-phistreep&*H012&*phi*pi*G/2)):

> difft012:=Map(x-> diff(x,t),t012);

difft012 :=[

0]

> t123:=evalm(-phistreep&*H123&*phi*pi*G/2):

> simplify(t123):

> t023:=simplify(evalm(-phistreep&*H023&*phi*pi*G/2));

t023 :=[ −2πG cos(t E + 2 px2 x+ 2 py2 y + 2 pz2 z)

]

> difft023:=Map(x-> diff(x,t),t023);

difft023 :=[

2πG sin(t E + 2 px2 x+ 2 py2 y + 2 pz2 z)E]

> t013:=simplify(evalm(-phistreep&*H013&*phi*pi*G/2));

t013 :=[

2πG sin(t E + 2 px2 x+ 2 py2 y + 2 pz2 z)]

> difft013:=Map(x-> diff(x,t),t013);

difft013 :=[

2πG cos(t E + 2 px2 x+ 2 py2 y + 2 pz2 z)E]